Dissection and area of composite figures

involving parts of a circle

Lesson 3

Learning outcomes

By the end of this lesson, students will be able to

dissect composite figures into at triangles, special quadrilaterals and parts of a circle
recognise and calculate the area of a composite figure as the sum of the areas of the dissected shapes (triangles, quadrilaterals and parts of a circle)

Content indicators

calculate the areas of composite figures by dissection into triangles, special quadrilaterals, quadrants, semicircles and sectors
Solve a variety of practical problems involving the areas of quadrilaterals and composite shapes

WM indicators

identify different possible dissections for a given composite figure and select an appropriate dissection to facilitate calculation of the area (Problem Solving)
calculate the area of an annulus (Problem Solving)

Prior knowledge and understanding

Students are assumed to be able to:

dissect irregular composite shapes into three or more triangles and special quadrilaterals
recall and use area formulas of circles and parts of a circle (including quadrants, semicircles)
find the area of composite shapes that are comprised multiple triangles and special quadrilaterals

Lesson Plan

Lesson length: 60 mins

Materials/Resources

Composite figure cutout (Introduction)
Composite figures (Body 1)
Modified tangram kit
Rulers
Compasses
Scissors
Butchers' paper
String
Magnets
Annulus activity worksheet

10:00 Introduction

Collect students' homework as students settle in class (AFL).

Check if completed/attempted
Assess if students understand how to dissect composite shapes and calculate their areas.

(7 min, LIT)

Recap how to dissect simple composite figures into triangles and special quadrilaterals
Review circle area formulas and how to calculate area of semicircles, quadrants and semicircles.

Provide students with the following composite figure. Ask (AFL):

What is a key difference between this composite figure and the ones we've seen previously?
What other basic shape do you think we will be including in our composite figures now?

Students should recognise that there is a curved edge, which means we may be using circles/circular parts.

Ask students to dissect the figure by drawing lines, as they have done in previous lessons, and to take particular care dissection of the curved section. Ask students to then hold up their dissections and check that students have dissected them appropriately (AFL). Students then paste their dissected figures into their workbooks.

Ask:

What shapes could we have dissected this into?
What shape(s) were used to create the curved surface?
If this shape did not have the curved surface, how would you calculate its area?
- Calculate the area underneath the semicircle (enclosed in red). (red A = 21 cm²)

Have students write their calculations in their workbook.

Ask students to recall the area formula for a circle (A = πr²) and how they will modify it to calculate the area of the semicircle (A = 1/2 x πr²). Have students then calculate the area of the semicircle, check that everyone has calculated it to be 28.27 cm²(2 dp).

Ask students to recall that to find the total area of a composite figure, they must sum the areas of the dissected components. Finally, check that all students calculated the total area to be 49.27 cm².

10:07 Body activity 1

(15 mins): Dissection of composite figures involving parts of a circle

In pairs, students dissect given outlines of two composite figures, printed on grid paper, created from a modified tangram template (to include curved shapes such as quadrants and semicircles). They are to find at least two different configurations to dissect the figure.

Modified tangram kit

Figure 1.

Figure 2.

Pairs calculate the area of the shapes by finding the area of individual components first, and then summing them together.

When students are working, circulate around the room and ask questions such as (AFL):

How many shapes are you cutting the figure into?
Is there another way to dissect the figure? Could you show me?
What is the area formula this shape? Can you show me the sides/dimensions you need to calculate the area?
How did you find the total area of your figure?

As in the previous lessons, students should recognise different ways of the same composite figures will result with the same calculated area.

10:21 Body activity 2

(28 mins): Calculating the area of an annulus (LIT)

(3 mins) Show students the following images of annuli. Ask students to identify the shapes present and their placement in relation to each other. Ask to ensure they know that the dot in the middle denotes the centre of both circles. (AFL - diagnostic assessment)

State that these are examples of a composite figure called an "annulus".

Annulus: plane figure that is the area between a pair of concentric circles
Concentric: used to describe shapes of different sizes, especially circles and arcs, that share the same centre

As a class discussion, ask:

Where can you find annuli in the real world?
- Responses may include: doughnuts, wheels, tree trunks, ripples, rings, etc.
How could we find the area of the shaded space of the between the two concentric circles of an annulus? (AFL)
- Mention that we will not sum areas like in the previous examples.

Segue into constructing and calculating the areas of annuli.

(10 mins) Model how to construct an annulus whilst students follow along on piece of coloured paper using a compass. (Do this by drawing a large scale version of the student activity using butcher's paper, string, magnets, dark maker etc.)

Instruct students to draw a circle with 5 cm radius in their on pieces of paper and label this C1. Ask them to find the area of this circle in their workbooks using the area formula for a circle (A = πr²) such that C1 = 25π cm²= 78.53 cm².
Using the same centre, construct another circle with radius 3 cm, as to create a pair of concentric circles. Label the smaller circle C2 and calculate its area in their books (C2 = 9π cm²= 28.27 cm²).
Carefully cut out C2 from the centre of the circle, keeping it intact.
Demonstrate that because we have taken out the middle area of the smaller circle, they should be able to recognise that the area of the 'ring' is found by subtracting C2 from C1 (i.e. A = C1 - C2 = 78.53 cm² - 28.27 cm²= 50.26 cm²).

From this, students should recognise that the area of an annulus is found by taking the area of the smaller circle from the area of the smaller circle. Hence, they need to calculate the difference between the areas of the two circles. Students paste their both paper cutouts in their workbooks.

3. C2 taken out of C1 to create an annulus.

1. Circle, C1 with R = 5 cm, has area 78.53 cm²

2. Concentric circle, C2 with r = 3 cm has area 28.27 cm²

As students are working, roam around the classroom to ensure student are on task and are calculating the areas correctly.

4. The area of the annulus, A, was calculated by finding the difference between C1 and C2.

(15 mins) Individually, students construct their own annuli and find their areas by following the instructions in the worksheet:

L3 - Area of an annulus

As students are working, ask (AFL):

How are the annuli in Part A and Part B related? How about in Part B and C?
What does a larger/smaller difference in the radii of two circles look like in the annulus? Describe how that might change the gap between the two rings.
- How do you think this affect the area of the annulus?
Does a larger outer ring always mean a larger annulus area? Why/why not? What else does this depend on? (Ext.)
Explain to me how the radii of two circles will affect the area of the annulus.

NB: Differentiation

Low-ability students may work in pairs to construct the annuli, calculate their areas and evaluate the differences
Higher-ability students could use their understanding of the circle area formulas and like terms to derive the following formula for the area of an annulus:
- A = π (R² – r²)

10:49 Conclusion

(6 mins) Class discussion: Discuss final answers to the worksheet. Ask (AFL):

How can we calculate the area of an annulus?
How were the annuli in Part A and Part B related? How about in Part B and C?
So what can we tell about the differences in radii and their effect on the look on the annulus?
How did we show this in our calculations? Did the area increase/decrease as the difference in radii became larger/smaller?
Does a larger outer ring always mean a larger annulus area? Why/why not? What else does this depend on?

Students should now know that the area of an annulus is calculated by finding by subtractingi the area of one circles from a larger, concentric one. They should also recognise that by increasing the difference between the radius of concentric circles, the area of the annuli also increases, and vice versa.

Also, reiterate that to calculate the area of composite figures involving parts of a circle follows the same process as composite figures with triangles and special quadrilaterals covered in Lessons 1 and 2.

(5 mins) Homework and pack up

Using the modified tangram kit, find two valid dissections of the following composite shape. Calculate the area for each dissection, showing full working out, and compare the values.
Find two examples of annuli in your home (e.g. roll of tape/ribbon, top of a can of food, stove top, etc.). Draw a sketch of the annulus, using a compass, record its radius and calculate its area.

Homework 1.

Evaluation

Did students have the relevant prior knowledge?
- Did they recall how to dissect and calculate the area of composite shapes?
- Did they recall the area formulas for circles, semicircles, quadrants and sectors?
- Review feedback from homework
Were my instructions clear?
Were all students engaged in the activities?
Were my questions clear/encouraged students to think about the concept?
Can students dissect a composite shape into smaller, basic shapes, including parts of a circle?
Can students find the area of an annulus?
What do I need to modify in my upcoming lessons to support the needs of all students?

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